{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 描述性统计 Statistical Summary"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 这项工作主要是让我们知道数据的整体状况怎么样，描述这个数据的“样子”。\n",
    "* 数据处理的最关键，也是最重要的第一步\n",
    "* 了解数据的概况，有助于后续的数据分析和挖掘"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**描述性统计的Python工具**\n",
    "> PANDAS\n",
    "\n",
    "> NumPy和SciPy\n",
    "* count 统计非NA的数量\n",
    "* describe 针对series或者DF的列计算汇总统计\n",
    "* min max 最小值和最大值\n",
    "* quantile 样本分位数\n",
    "* sum 求和\n",
    "* mean 均值\n",
    "* median 中位数\n",
    "* mad 根据均值计算平均绝对离差\n",
    "* var 方差\n",
    "* std 标准差\n",
    "* skew 偏度（三阶矩）\n",
    "* kurt 峰度\n",
    "* cumsum 累计和\n",
    "* cumprod 累计乘积\n",
    "* diff 一阶差分\n",
    "* pct_change 计算百分数变化\n",
    "* mode 计算众数\n",
    "* cov 协方差\n",
    "* corrcoef 相关系数"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np \n",
    "import pandas as pd \n",
    "import scipy.stats as stats \n",
    "import scipy\n",
    "\n",
    "from datetime import datetime \n",
    "import statsmodels.formula.api as smf  # OLS regression\n",
    "\n",
    "from matplotlib import style\n",
    "import matplotlib.pyplot as plt \n",
    "import matplotlib.dates as mdates\n",
    "\n",
    "from matplotlib.font_manager import FontProperties # 作图中文\n",
    "from pylab import mpl\n",
    "#mpl.rcParams['font.sans-serif'] = ['SimHei']\n",
    "#plt.rcParams['font.family'] = 'Times New Roman'\n",
    "\n",
    "#输出矢量图 渲染矢量图\n",
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format = 'svg'\n",
    "\n",
    "from IPython.core.interactiveshell import InteractiveShell # jupyter运行输出的模块\n",
    "#显示每一个运行结果\n",
    "InteractiveShell.ast_node_interactivity = 'all'\n",
    "\n",
    "#设置行不限制数量\n",
    "#pd.set_option('display.max_rows',None)\n",
    "\n",
    "#设置列不限制数量\n",
    "pd.set_option('display.max_columns', None)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.microsoft.datawrangler.viewer.v0+json": {
       "columns": [
        {
         "name": "Day",
         "rawType": "datetime64[ns]",
         "type": "datetime"
        },
        {
         "name": "Preclose",
         "rawType": "object",
         "type": "string"
        },
        {
         "name": "Open",
         "rawType": "float64",
         "type": "float"
        },
        {
         "name": "Highest",
         "rawType": "float64",
         "type": "float"
        },
        {
         "name": "Lowest",
         "rawType": "float64",
         "type": "float"
        },
        {
         "name": "Close",
         "rawType": "float64",
         "type": "float"
        }
       ],
       "ref": "617d440a-278d-46ec-a638-464560bb0f71",
       "rows": [
        [
         "1990-12-19 00:00:00",
         "        ",
         "96.05",
         "99.98",
         "95.79",
         "99.98"
        ],
        [
         "1990-12-20 00:00:00",
         "99.98",
         "104.3",
         "104.39",
         "99.98",
         "104.39"
        ],
        [
         "1990-12-21 00:00:00",
         "104.39",
         "109.07",
         "109.13",
         "103.73",
         "109.13"
        ],
        [
         "1990-12-24 00:00:00",
         "109.13",
         "113.57",
         "114.55",
         "109.13",
         "114.55"
        ],
        [
         "1990-12-25 00:00:00",
         "114.55",
         "120.09",
         "120.25",
         "114.55",
         "120.25"
        ],
        [
         "1990-12-26 00:00:00",
         "120.25",
         "125.27",
         "125.27",
         "120.25",
         "125.27"
        ],
        [
         "1990-12-27 00:00:00",
         "125.27",
         "125.27",
         "125.28",
         "125.27",
         "125.28"
        ],
        [
         "1990-12-28 00:00:00",
         "125.28",
         "126.39",
         "126.45",
         "125.28",
         "126.45"
        ],
        [
         "1990-12-31 00:00:00",
         "126.45",
         "126.56",
         "127.61",
         "126.48",
         "127.61"
        ],
        [
         "1991-01-02 00:00:00",
         "127.61",
         "127.61",
         "128.84",
         "127.61",
         "128.84"
        ],
        [
         "1991-01-03 00:00:00",
         "128.84",
         "128.84",
         "130.14",
         "128.84",
         "130.14"
        ],
        [
         "1991-01-04 00:00:00",
         "130.14",
         "131.27",
         "131.44",
         "130.14",
         "131.44"
        ],
        [
         "1991-01-07 00:00:00",
         "131.44",
         "131.99",
         "132.06",
         "131.45",
         "132.06"
        ],
        [
         "1991-01-08 00:00:00",
         "132.06",
         "132.62",
         "132.68",
         "132.06",
         "132.68"
        ],
        [
         "1991-01-09 00:00:00",
         "132.68",
         "133.3",
         "133.34",
         "132.68",
         "133.34"
        ],
        [
         "1991-01-10 00:00:00",
         "133.34",
         "133.93",
         "133.97",
         "133.34",
         "133.97"
        ],
        [
         "1991-01-11 00:00:00",
         "133.97",
         "134.61",
         "134.61",
         "134.51",
         "134.6"
        ],
        [
         "1991-01-14 00:00:00",
         "134.6",
         "134.11",
         "135.19",
         "134.11",
         "134.67"
        ],
        [
         "1991-01-15 00:00:00",
         "134.67",
         "134.21",
         "134.74",
         "134.19",
         "134.74"
        ],
        [
         "1991-01-16 00:00:00",
         "134.74",
         "134.19",
         "134.74",
         "134.14",
         "134.24"
        ],
        [
         "1991-01-17 00:00:00",
         "134.24",
         "133.67",
         "134.25",
         "133.65",
         "134.25"
        ],
        [
         "1991-01-18 00:00:00",
         "134.25",
         "133.7",
         "134.25",
         "133.67",
         "134.24"
        ],
        [
         "1991-01-21 00:00:00",
         "134.24",
         "133.7",
         "134.24",
         "133.66",
         "134.24"
        ],
        [
         "1991-01-22 00:00:00",
         "134.24",
         "133.72",
         "134.24",
         "133.66",
         "133.72"
        ],
        [
         "1991-01-23 00:00:00",
         "133.72",
         "133.17",
         "133.72",
         "133.14",
         "133.17"
        ],
        [
         "1991-01-24 00:00:00",
         "133.17",
         "132.61",
         "133.17",
         "132.57",
         "132.61"
        ],
        [
         "1991-01-25 00:00:00",
         "132.61",
         "132.05",
         "132.07",
         "132.03",
         "132.05"
        ],
        [
         "1991-01-28 00:00:00",
         "132.05",
         "131.46",
         "131.55",
         "131.46",
         "131.46"
        ],
        [
         "1991-01-29 00:00:00",
         "131.46",
         "130.95",
         "130.97",
         "130.95",
         "130.95"
        ],
        [
         "1991-01-30 00:00:00",
         "130.95",
         "130.44",
         "130.95",
         "130.41",
         "130.44"
        ],
        [
         "1991-01-31 00:00:00",
         "130.44",
         "129.93",
         "130.46",
         "129.93",
         "129.97"
        ],
        [
         "1991-02-01 00:00:00",
         "129.97",
         "129.5",
         "129.97",
         "129.45",
         "129.51"
        ],
        [
         "1991-02-04 00:00:00",
         "129.51",
         "129.05",
         "129.58",
         "129.05",
         "129.05"
        ],
        [
         "1991-02-05 00:00:00",
         "129.05",
         "128.56",
         "128.58",
         "128.53",
         "128.58"
        ],
        [
         "1991-02-06 00:00:00",
         "128.58",
         "129.13",
         "129.15",
         "128.06",
         "129.14"
        ],
        [
         "1991-02-07 00:00:00",
         "129.14",
         "129.74",
         "129.79",
         "129.14",
         "129.79"
        ],
        [
         "1991-02-08 00:00:00",
         "129.79",
         "130.36",
         "130.39",
         "129.79",
         "130.38"
        ],
        [
         "1991-02-11 00:00:00",
         "130.38",
         "130.92",
         "130.97",
         "130.39",
         "130.97"
        ],
        [
         "1991-02-12 00:00:00",
         "130.97",
         "131.54",
         "131.56",
         "130.97",
         "131.35"
        ],
        [
         "1991-02-13 00:00:00",
         "131.35",
         "131.93",
         "131.93",
         "131.35",
         "131.92"
        ],
        [
         "1991-02-14 00:00:00",
         "131.92",
         "132.53",
         "132.53",
         "132.3",
         "132.53"
        ],
        [
         "1991-02-19 00:00:00",
         "132.53",
         "133.12",
         "133.14",
         "133.08",
         "133.13"
        ],
        [
         "1991-02-20 00:00:00",
         "133.13",
         "133.63",
         "133.67",
         "133.13",
         "133.67"
        ],
        [
         "1991-02-21 00:00:00",
         "133.67",
         "134.24",
         "134.28",
         "133.67",
         "134.28"
        ],
        [
         "1991-02-22 00:00:00",
         "134.28",
         "134.85",
         "134.87",
         "134.28",
         "134.87"
        ],
        [
         "1991-02-25 00:00:00",
         "134.87",
         "134.37",
         "134.87",
         "134.33",
         "134.4"
        ],
        [
         "1991-02-26 00:00:00",
         "134.4",
         "133.9",
         "134.44",
         "133.9",
         "133.93"
        ],
        [
         "1991-02-27 00:00:00",
         "133.93",
         "133.44",
         "133.98",
         "133.44",
         "133.47"
        ],
        [
         "1991-02-28 00:00:00",
         "133.47",
         "132.99",
         "133.52",
         "132.98",
         "133.01"
        ],
        [
         "1991-03-01 00:00:00",
         "133.01",
         "132.53",
         "132.53",
         "132.47",
         "132.53"
        ]
       ],
       "shape": {
        "columns": 5,
        "rows": 8473
       }
      },
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Preclose</th>\n",
       "      <th>Open</th>\n",
       "      <th>Highest</th>\n",
       "      <th>Lowest</th>\n",
       "      <th>Close</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Day</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1990-12-19</th>\n",
       "      <td></td>\n",
       "      <td>96.050</td>\n",
       "      <td>99.980</td>\n",
       "      <td>95.790</td>\n",
       "      <td>99.980</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1990-12-20</th>\n",
       "      <td>99.98</td>\n",
       "      <td>104.300</td>\n",
       "      <td>104.390</td>\n",
       "      <td>99.980</td>\n",
       "      <td>104.390</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1990-12-21</th>\n",
       "      <td>104.39</td>\n",
       "      <td>109.070</td>\n",
       "      <td>109.130</td>\n",
       "      <td>103.730</td>\n",
       "      <td>109.130</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1990-12-24</th>\n",
       "      <td>109.13</td>\n",
       "      <td>113.570</td>\n",
       "      <td>114.550</td>\n",
       "      <td>109.130</td>\n",
       "      <td>114.550</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1990-12-25</th>\n",
       "      <td>114.55</td>\n",
       "      <td>120.090</td>\n",
       "      <td>120.250</td>\n",
       "      <td>114.550</td>\n",
       "      <td>120.250</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2025-08-25</th>\n",
       "      <td>3825.759</td>\n",
       "      <td>3848.163</td>\n",
       "      <td>3883.562</td>\n",
       "      <td>3839.972</td>\n",
       "      <td>3883.562</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2025-08-26</th>\n",
       "      <td>3883.562</td>\n",
       "      <td>3871.471</td>\n",
       "      <td>3888.599</td>\n",
       "      <td>3859.758</td>\n",
       "      <td>3868.382</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2025-08-27</th>\n",
       "      <td>3868.382</td>\n",
       "      <td>3869.612</td>\n",
       "      <td>3887.198</td>\n",
       "      <td>3800.350</td>\n",
       "      <td>3800.350</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2025-08-28</th>\n",
       "      <td>3800.35</td>\n",
       "      <td>3796.711</td>\n",
       "      <td>3845.087</td>\n",
       "      <td>3761.422</td>\n",
       "      <td>3843.597</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2025-08-29</th>\n",
       "      <td>3843.597</td>\n",
       "      <td>3842.823</td>\n",
       "      <td>3867.606</td>\n",
       "      <td>3839.206</td>\n",
       "      <td>3857.927</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>8473 rows × 5 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "            Preclose      Open   Highest    Lowest     Close\n",
       "Day                                                         \n",
       "1990-12-19              96.050    99.980    95.790    99.980\n",
       "1990-12-20     99.98   104.300   104.390    99.980   104.390\n",
       "1990-12-21    104.39   109.070   109.130   103.730   109.130\n",
       "1990-12-24    109.13   113.570   114.550   109.130   114.550\n",
       "1990-12-25    114.55   120.090   120.250   114.550   120.250\n",
       "...              ...       ...       ...       ...       ...\n",
       "2025-08-25  3825.759  3848.163  3883.562  3839.972  3883.562\n",
       "2025-08-26  3883.562  3871.471  3888.599  3859.758  3868.382\n",
       "2025-08-27  3868.382  3869.612  3887.198  3800.350  3800.350\n",
       "2025-08-28   3800.35  3796.711  3845.087  3761.422  3843.597\n",
       "2025-08-29  3843.597  3842.823  3867.606  3839.206  3857.927\n",
       "\n",
       "[8473 rows x 5 columns]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = pd.read_csv('datasets/000001.csv')\n",
    "data['Day'] = pd.to_datetime(data['Day'],format='%Y/%m/%d')\n",
    "data.set_index('Day', inplace = True)\n",
    "data.sort_values(by = ['Day'], ascending=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.microsoft.datawrangler.viewer.v0+json": {
       "columns": [
        {
         "name": "Day",
         "rawType": "datetime64[ns]",
         "type": "datetime"
        },
        {
         "name": "Preclose",
         "rawType": "float64",
         "type": "float"
        },
        {
         "name": "Open",
         "rawType": "float64",
         "type": "float"
        },
        {
         "name": "Highest",
         "rawType": "float64",
         "type": "float"
        },
        {
         "name": "Lowest",
         "rawType": "float64",
         "type": "float"
        },
        {
         "name": "Close",
         "rawType": "float64",
         "type": "float"
        },
        {
         "name": "Raw_return",
         "rawType": "float64",
         "type": "float"
        }
       ],
       "ref": "11fa1384-c08a-4343-a0e5-4c187d0a81fd",
       "rows": [
        [
         "1995-01-03 00:00:00",
         "647.87",
         "637.72",
         "647.71",
         "630.53",
         "639.88",
         "-0.01233272107058514"
        ],
        [
         "1995-01-04 00:00:00",
         "639.88",
         "641.9",
         "655.51",
         "638.86",
         "653.81",
         "0.021769706820028656"
        ],
        [
         "1995-01-05 00:00:00",
         "653.81",
         "655.38",
         "657.52",
         "645.81",
         "646.89",
         "-0.010584114651045362"
        ],
        [
         "1995-01-06 00:00:00",
         "646.89",
         "642.75",
         "643.89",
         "636.33",
         "640.76",
         "-0.00947610876656002"
        ],
        [
         "1995-01-09 00:00:00",
         "640.76",
         "637.52",
         "637.55",
         "625.04",
         "626.0",
         "-0.023035145764404796"
        ],
        [
         "1995-01-10 00:00:00",
         "626.0",
         "616.16",
         "617.6",
         "607.06",
         "610.3",
         "-0.02507987220447294"
        ],
        [
         "1995-01-11 00:00:00",
         "610.3",
         "610.3",
         "615.21",
         "606.92",
         "613.13",
         "0.004637063739144853"
        ],
        [
         "1995-01-12 00:00:00",
         "613.13",
         "612.79",
         "615.47",
         "601.17",
         "602.12",
         "-0.017957040105687216"
        ],
        [
         "1995-01-13 00:00:00",
         "602.12",
         "597.32",
         "601.07",
         "593.81",
         "597.84",
         "-0.007108217631036973"
        ],
        [
         "1995-01-16 00:00:00",
         "597.84",
         "591.06",
         "591.32",
         "584.88",
         "588.81",
         "-0.015104375752709953"
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       "<div>\n",
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Preclose</th>\n",
       "      <th>Open</th>\n",
       "      <th>Highest</th>\n",
       "      <th>Lowest</th>\n",
       "      <th>Close</th>\n",
       "      <th>Raw_return</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Day</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1995-01-03</th>\n",
       "      <td>647.870</td>\n",
       "      <td>637.720</td>\n",
       "      <td>647.710</td>\n",
       "      <td>630.530</td>\n",
       "      <td>639.880</td>\n",
       "      <td>-0.012333</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1995-01-04</th>\n",
       "      <td>639.880</td>\n",
       "      <td>641.900</td>\n",
       "      <td>655.510</td>\n",
       "      <td>638.860</td>\n",
       "      <td>653.810</td>\n",
       "      <td>0.021770</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1995-01-05</th>\n",
       "      <td>653.810</td>\n",
       "      <td>655.380</td>\n",
       "      <td>657.520</td>\n",
       "      <td>645.810</td>\n",
       "      <td>646.890</td>\n",
       "      <td>-0.010584</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1995-01-06</th>\n",
       "      <td>646.890</td>\n",
       "      <td>642.750</td>\n",
       "      <td>643.890</td>\n",
       "      <td>636.330</td>\n",
       "      <td>640.760</td>\n",
       "      <td>-0.009476</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1995-01-09</th>\n",
       "      <td>640.760</td>\n",
       "      <td>637.520</td>\n",
       "      <td>637.550</td>\n",
       "      <td>625.040</td>\n",
       "      <td>626.000</td>\n",
       "      <td>-0.023035</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2025-08-25</th>\n",
       "      <td>3825.759</td>\n",
       "      <td>3848.163</td>\n",
       "      <td>3883.562</td>\n",
       "      <td>3839.972</td>\n",
       "      <td>3883.562</td>\n",
       "      <td>0.015109</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2025-08-26</th>\n",
       "      <td>3883.562</td>\n",
       "      <td>3871.471</td>\n",
       "      <td>3888.599</td>\n",
       "      <td>3859.758</td>\n",
       "      <td>3868.382</td>\n",
       "      <td>-0.003909</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2025-08-27</th>\n",
       "      <td>3868.382</td>\n",
       "      <td>3869.612</td>\n",
       "      <td>3887.198</td>\n",
       "      <td>3800.350</td>\n",
       "      <td>3800.350</td>\n",
       "      <td>-0.017587</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2025-08-28</th>\n",
       "      <td>3800.350</td>\n",
       "      <td>3796.711</td>\n",
       "      <td>3845.087</td>\n",
       "      <td>3761.422</td>\n",
       "      <td>3843.597</td>\n",
       "      <td>0.011380</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2025-08-29</th>\n",
       "      <td>3843.597</td>\n",
       "      <td>3842.823</td>\n",
       "      <td>3867.606</td>\n",
       "      <td>3839.206</td>\n",
       "      <td>3857.927</td>\n",
       "      <td>0.003728</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>7445 rows × 6 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "            Preclose      Open   Highest    Lowest     Close  Raw_return\n",
       "Day                                                                     \n",
       "1995-01-03   647.870   637.720   647.710   630.530   639.880   -0.012333\n",
       "1995-01-04   639.880   641.900   655.510   638.860   653.810    0.021770\n",
       "1995-01-05   653.810   655.380   657.520   645.810   646.890   -0.010584\n",
       "1995-01-06   646.890   642.750   643.890   636.330   640.760   -0.009476\n",
       "1995-01-09   640.760   637.520   637.550   625.040   626.000   -0.023035\n",
       "...              ...       ...       ...       ...       ...         ...\n",
       "2025-08-25  3825.759  3848.163  3883.562  3839.972  3883.562    0.015109\n",
       "2025-08-26  3883.562  3871.471  3888.599  3859.758  3868.382   -0.003909\n",
       "2025-08-27  3868.382  3869.612  3887.198  3800.350  3800.350   -0.017587\n",
       "2025-08-28  3800.350  3796.711  3845.087  3761.422  3843.597    0.011380\n",
       "2025-08-29  3843.597  3842.823  3867.606  3839.206  3857.927    0.003728\n",
       "\n",
       "[7445 rows x 6 columns]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data_new = data['1995-01':'2025-08'].copy()\n",
    "data_new['Close'] = pd.to_numeric(data_new['Close'])\n",
    "data_new['Preclose'] = pd.to_numeric(data_new['Preclose'])\n",
    "# 计算000001上证指数日收益率 两种：\n",
    "data_new['Raw_return'] = data_new['Close'] / data_new['Preclose'] - 1\n",
    "data_new"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
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       "      <th></th>\n",
       "      <th>Ret</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Day</th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1995-01-31</th>\n",
       "      <td>-0.131631</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1995-02-28</th>\n",
       "      <td>-0.023694</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1995-03-31</th>\n",
       "      <td>0.177803</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1995-04-30</th>\n",
       "      <td>-0.103552</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1995-05-31</th>\n",
       "      <td>0.207922</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2025-04-30</th>\n",
       "      <td>-0.017002</td>\n",
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       "    <tr>\n",
       "      <th>2025-05-31</th>\n",
       "      <td>0.020877</td>\n",
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       "    <tr>\n",
       "      <th>2025-06-30</th>\n",
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       "    <tr>\n",
       "      <th>2025-07-31</th>\n",
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       "    <tr>\n",
       "      <th>2025-08-31</th>\n",
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       "  </tbody>\n",
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       "<p>368 rows × 1 columns</p>\n",
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      ],
      "text/plain": [
       "                 Ret\n",
       "Day                 \n",
       "1995-01-31 -0.131631\n",
       "1995-02-28 -0.023694\n",
       "1995-03-31  0.177803\n",
       "1995-04-30 -0.103552\n",
       "1995-05-31  0.207922\n",
       "...              ...\n",
       "2025-04-30 -0.017002\n",
       "2025-05-31  0.020877\n",
       "2025-06-30  0.028959\n",
       "2025-07-31  0.037389\n",
       "2025-08-31  0.079682\n",
       "\n",
       "[368 rows x 1 columns]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Month_data = data_new.resample('ME')['Raw_return'].apply(lambda x: (1+x).prod()-1).to_frame(name = 'Ret')\n",
    "Month_data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
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         "1998-12-31 00:00:00",
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       "      <th>1995-01-31</th>\n",
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       "      <th>1995-02-28</th>\n",
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       "                 Ret\n",
       "month               \n",
       "1995-01-31 -0.131631\n",
       "1995-02-28 -0.023694\n",
       "1995-03-31  0.177803\n",
       "1995-04-30 -0.103552\n",
       "1995-05-31  0.207922\n",
       "...              ...\n",
       "2025-04-30 -0.017002\n",
       "2025-05-31  0.020877\n",
       "2025-06-30  0.028959\n",
       "2025-07-31  0.037389\n",
       "2025-08-31  0.079682\n",
       "\n",
       "[368 rows x 1 columns]"
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    "Month_data = data_new.resample('ME')['Raw_return'].apply(lambda x: np.prod(1+x)-1).to_frame(name = 'Ret')\n",
    "Month_data.index.name = 'month'\n",
    "Month_data"
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   "metadata": {},
   "outputs": [
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      "text/plain": [
       "                 Ret\n",
       "Q                   \n",
       "1995-03-31 -0.001466\n",
       "1995-06-30 -0.025258\n",
       "1995-09-30  0.145660\n",
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       "\n",
       "[123 rows x 1 columns]"
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     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "Quarter_data = data_new.resample('QE')['Raw_return'].apply(lambda x: np.prod(1+x)-1).to_frame(name = 'Ret')\n",
    "Quarter_data.index.name = 'Q'\n",
    "Quarter_data"
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  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
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        {
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       "rows": [
        [
         "1995-12-31 00:00:00",
         "-0.1428990383873303"
        ],
        [
         "1996-12-31 00:00:00",
         "0.6514253813322783"
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        [
         "1997-12-31 00:00:00",
         "0.3021526248064368"
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        [
         "1998-12-31 00:00:00",
         "-0.03969516790888461"
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         "1999-12-31 00:00:00",
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         "2000-12-31 00:00:00",
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         "2003-12-31 00:00:00",
         "0.10267005487423031"
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        [
         "2004-12-31 00:00:00",
         "-0.1539972211831364"
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        [
         "2005-12-31 00:00:00",
         "-0.08325305961310714"
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        [
         "2006-12-31 00:00:00",
         "1.3043339706819643"
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        [
         "2007-12-31 00:00:00",
         "0.9665927855666503"
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        [
         "2008-12-31 00:00:00",
         "-0.6539410364986822"
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        [
         "2009-12-31 00:00:00",
         "0.7998253524530277"
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        [
         "2010-12-31 00:00:00",
         "-0.14313090072441115"
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        [
         "2011-12-31 00:00:00",
         "-0.21675308395772197"
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        [
         "2012-12-31 00:00:00",
         "0.03169471951696434"
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        [
         "2013-12-31 00:00:00",
         "-0.06749282764760167"
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        [
         "2014-12-31 00:00:00",
         "0.5286911974593327"
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        [
         "2015-12-31 00:00:00",
         "0.09413605055214114"
        ],
        [
         "2016-12-31 00:00:00",
         "-0.12306240428573822"
        ],
        [
         "2017-12-31 00:00:00",
         "0.06557783763580938"
        ],
        [
         "2018-12-31 00:00:00",
         "-0.24591115666869323"
        ],
        [
         "2019-12-31 00:00:00",
         "0.22303219856449563"
        ],
        [
         "2020-12-31 00:00:00",
         "0.1386666754094923"
        ],
        [
         "2021-12-31 00:00:00",
         "0.04800076013440857"
        ],
        [
         "2022-12-31 00:00:00",
         "-0.1512503907782794"
        ],
        [
         "2023-12-31 00:00:00",
         "-0.03700665457551999"
        ],
        [
         "2024-12-31 00:00:00",
         "0.12666771722190373"
        ],
        [
         "2025-12-31 00:00:00",
         "0.15101425727296225"
        ]
       ],
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Ret</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Y</th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1995-12-31</th>\n",
       "      <td>-0.142899</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1996-12-31</th>\n",
       "      <td>0.651425</td>\n",
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       "    <tr>\n",
       "      <th>1997-12-31</th>\n",
       "      <td>0.302153</td>\n",
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       "    <tr>\n",
       "      <th>1998-12-31</th>\n",
       "      <td>-0.039695</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1999-12-31</th>\n",
       "      <td>0.191750</td>\n",
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       "    <tr>\n",
       "      <th>2000-12-31</th>\n",
       "      <td>0.517277</td>\n",
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       "      <th>2001-12-31</th>\n",
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       "    <tr>\n",
       "      <th>2002-12-31</th>\n",
       "      <td>-0.175167</td>\n",
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       "    <tr>\n",
       "      <th>2003-12-31</th>\n",
       "      <td>0.102670</td>\n",
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       "      <th>2004-12-31</th>\n",
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       "      <th>2005-12-31</th>\n",
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       "      <th>2006-12-31</th>\n",
       "      <td>1.304334</td>\n",
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       "      <th>2007-12-31</th>\n",
       "      <td>0.966593</td>\n",
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       "    <tr>\n",
       "      <th>2008-12-31</th>\n",
       "      <td>-0.653941</td>\n",
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       "    <tr>\n",
       "      <th>2009-12-31</th>\n",
       "      <td>0.799825</td>\n",
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       "    <tr>\n",
       "      <th>2010-12-31</th>\n",
       "      <td>-0.143131</td>\n",
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       "    <tr>\n",
       "      <th>2011-12-31</th>\n",
       "      <td>-0.216753</td>\n",
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       "    <tr>\n",
       "      <th>2012-12-31</th>\n",
       "      <td>0.031695</td>\n",
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       "    <tr>\n",
       "      <th>2013-12-31</th>\n",
       "      <td>-0.067493</td>\n",
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       "    <tr>\n",
       "      <th>2014-12-31</th>\n",
       "      <td>0.528691</td>\n",
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       "    <tr>\n",
       "      <th>2015-12-31</th>\n",
       "      <td>0.094136</td>\n",
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       "    <tr>\n",
       "      <th>2016-12-31</th>\n",
       "      <td>-0.123062</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2017-12-31</th>\n",
       "      <td>0.065578</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2018-12-31</th>\n",
       "      <td>-0.245911</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2019-12-31</th>\n",
       "      <td>0.223032</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2020-12-31</th>\n",
       "      <td>0.138667</td>\n",
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       "    <tr>\n",
       "      <th>2021-12-31</th>\n",
       "      <td>0.048001</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2022-12-31</th>\n",
       "      <td>-0.151250</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2023-12-31</th>\n",
       "      <td>-0.037007</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2024-12-31</th>\n",
       "      <td>0.126668</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2025-12-31</th>\n",
       "      <td>0.151014</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                 Ret\n",
       "Y                   \n",
       "1995-12-31 -0.142899\n",
       "1996-12-31  0.651425\n",
       "1997-12-31  0.302153\n",
       "1998-12-31 -0.039695\n",
       "1999-12-31  0.191750\n",
       "2000-12-31  0.517277\n",
       "2001-12-31 -0.206180\n",
       "2002-12-31 -0.175167\n",
       "2003-12-31  0.102670\n",
       "2004-12-31 -0.153997\n",
       "2005-12-31 -0.083253\n",
       "2006-12-31  1.304334\n",
       "2007-12-31  0.966593\n",
       "2008-12-31 -0.653941\n",
       "2009-12-31  0.799825\n",
       "2010-12-31 -0.143131\n",
       "2011-12-31 -0.216753\n",
       "2012-12-31  0.031695\n",
       "2013-12-31 -0.067493\n",
       "2014-12-31  0.528691\n",
       "2015-12-31  0.094136\n",
       "2016-12-31 -0.123062\n",
       "2017-12-31  0.065578\n",
       "2018-12-31 -0.245911\n",
       "2019-12-31  0.223032\n",
       "2020-12-31  0.138667\n",
       "2021-12-31  0.048001\n",
       "2022-12-31 -0.151250\n",
       "2023-12-31 -0.037007\n",
       "2024-12-31  0.126668\n",
       "2025-12-31  0.151014"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Year_data = data_new.resample('YE')['Raw_return'].apply(lambda x: np.prod(1+x)-1).to_frame(name = 'Ret')\n",
    "Year_data.index.name = 'Y'\n",
    "Year_data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
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        {
         "name": "index",
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         "type": "integer"
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         "rawType": "float64",
         "type": "float"
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         "name": "最小值",
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         "name": "25%分位数",
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         "name": "75%分位数",
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         "name": "最大值",
         "rawType": "float64",
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         "name": "正收益比例(%)",
         "rawType": "float64",
         "type": "float"
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         "name": "峰度",
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       "      <th>样本数</th>\n",
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       "      <th>中位数</th>\n",
       "      <th>标准差</th>\n",
       "      <th>方差</th>\n",
       "      <th>最小值</th>\n",
       "      <th>25%分位数</th>\n",
       "      <th>50%分位数</th>\n",
       "      <th>75%分位数</th>\n",
       "      <th>最大值</th>\n",
       "      <th>正收益比例(%)</th>\n",
       "      <th>偏度</th>\n",
       "      <th>峰度</th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>日收益率</th>\n",
       "      <td>6219</td>\n",
       "      <td>0.00027</td>\n",
       "      <td>0.00051</td>\n",
       "      <td>0.01462</td>\n",
       "      <td>0.000214</td>\n",
       "      <td>-0.08841</td>\n",
       "      <td>-0.00615</td>\n",
       "      <td>0.00051</td>\n",
       "      <td>0.00700</td>\n",
       "      <td>0.09857</td>\n",
       "      <td>52.68</td>\n",
       "      <td>-0.212</td>\n",
       "      <td>5.631</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>月收益率</th>\n",
       "      <td>308</td>\n",
       "      <td>0.00582</td>\n",
       "      <td>0.00457</td>\n",
       "      <td>0.06979</td>\n",
       "      <td>0.004870</td>\n",
       "      <td>-0.24631</td>\n",
       "      <td>-0.03590</td>\n",
       "      <td>0.00457</td>\n",
       "      <td>0.04199</td>\n",
       "      <td>0.27446</td>\n",
       "      <td>54.22</td>\n",
       "      <td>-0.055</td>\n",
       "      <td>1.991</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>年收益率</th>\n",
       "      <td>26</td>\n",
       "      <td>0.10927</td>\n",
       "      <td>0.03985</td>\n",
       "      <td>0.41714</td>\n",
       "      <td>0.174006</td>\n",
       "      <td>-0.65394</td>\n",
       "      <td>-0.14922</td>\n",
       "      <td>0.03985</td>\n",
       "      <td>0.14793</td>\n",
       "      <td>1.30433</td>\n",
       "      <td>53.85</td>\n",
       "      <td>1.284</td>\n",
       "      <td>2.094</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       样本数       均值      中位数      标准差        方差      最小值   25%分位数   50%分位数  \\\n",
       "日收益率  6219  0.00027  0.00051  0.01462  0.000214 -0.08841 -0.00615  0.00051   \n",
       "月收益率   308  0.00582  0.00457  0.06979  0.004870 -0.24631 -0.03590  0.00457   \n",
       "年收益率    26  0.10927  0.03985  0.41714  0.174006 -0.65394 -0.14922  0.03985   \n",
       "\n",
       "       75%分位数      最大值  正收益比例(%)     偏度     峰度  \n",
       "日收益率  0.00700  0.09857     52.68 -0.212  5.631  \n",
       "月收益率  0.04199  0.27446     54.22 -0.055  1.991  \n",
       "年收益率  0.14793  1.30433     53.85  1.284  2.094  "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 汇总日/月/年收益率的描述性统计指标\n",
    "def summarize_returns(series):\n",
    "    clean_series = series.dropna()\n",
    "    return pd.Series({\n",
    "        '样本数': clean_series.count(),\n",
    "        '均值': clean_series.mean(),\n",
    "        '中位数': clean_series.median(),\n",
    "        '标准差': clean_series.std(),\n",
    "        '方差': clean_series.var(),\n",
    "        '最小值': clean_series.min(),\n",
    "        '25%分位数': clean_series.quantile(0.25),\n",
    "        '50%分位数': clean_series.quantile(0.5),\n",
    "        '75%分位数': clean_series.quantile(0.75),\n",
    "        '最大值': clean_series.max(),\n",
    "        '正收益比例': (clean_series > 0).mean(),\n",
    "        '偏度': clean_series.skew(),\n",
    "        '峰度': clean_series.kurt()\n",
    "    })\n",
    "\n",
    "return_periods = {\n",
    "    '日收益率': data_new.loc['2000-01':'2025-08', 'Raw_return'],\n",
    "    '月收益率': Month_data.loc['2000-01':'2025-08', 'Ret'],\n",
    "    '年收益率': Year_data.loc['2000':'2025', 'Ret']\n",
    "}\n",
    "\n",
    "returns_summary = pd.DataFrame({\n",
    "    name: summarize_returns(series) for name, series in return_periods.items()\n",
    "}).T\n",
    "returns_summary['正收益比例'] = returns_summary['正收益比例'].mul(100)\n",
    "returns_summary.rename(columns={'正收益比例': '正收益比例(%)'}, inplace=True)\n",
    "returns_summary = returns_summary.round({\n",
    "    '样本数': 0,\n",
    "    '均值': 5,\n",
    "    '中位数': 5,\n",
    "    '标准差': 5,\n",
    "    '方差': 7,\n",
    "    '最小值': 5,\n",
    "    '25%分位数': 5,\n",
    "    '50%分位数': 5,\n",
    "    '75%分位数': 5,\n",
    "    '最大值': 5,\n",
    "    '正收益比例(%)': 2,\n",
    "    '偏度': 3,\n",
    "    '峰度': 3\n",
    "})\n",
    "returns_summary['样本数'] = returns_summary['样本数'].astype(int)\n",
    "returns_summary"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 均值 mean"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "算数平均：\n",
    "$$A_n=\\frac{a_1+a_2+a_3+\\cdots+a_n}{n}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(0.00026)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "round(data_new['2000-01':'2024-09']['Raw_return'].mean(),5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(0.0002576419497498928)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.mean(data_new['2000-01':'2024-09']['Raw_return'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "中国股票市场日度平均收益率为0.027%\n"
     ]
    }
   ],
   "source": [
    "print('中国股票市场日度平均收益率为',data_new['2000-01':'2025-08']['Raw_return'].mean().round(5)*100,'%',sep=\"\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "中国股票市场月度平均收益率为0.6%\n"
     ]
    }
   ],
   "source": [
    "print('中国股票市场月度平均收益率为',Month_data['2000-01':'2025-08']['Ret'].mean().round(3)*100,'%',sep=\"\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "中国股票市场年度平均收益率为10.927000000000001%\n"
     ]
    }
   ],
   "source": [
    "print('中国股票市场年度平均收益率为',Year_data['2000':'2025']['Ret'].mean().round(5)*100,'%',sep=\"\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0002576419497498928"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(data_new['2000-01':'2024-09']['Raw_return']) / len(data_new['2000-01':'2024-09']['Raw_return'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.microsoft.datawrangler.viewer.v0+json": {
       "columns": [
        {
         "name": "index",
         "rawType": "object",
         "type": "string"
        },
        {
         "name": "Raw_return",
         "rawType": "float64",
         "type": "float"
        }
       ],
       "ref": "68acfb94-56ab-4481-b26d-aec1cbd275fe",
       "rows": [
        [
         "count",
         "7223.0"
        ],
        [
         "mean",
         "0.0003617740741346291"
        ],
        [
         "std",
         "0.016463895699981516"
        ],
        [
         "min",
         "-0.16393661830580997"
        ],
        [
         "25%",
         "-0.006719339051194095"
        ],
        [
         "50%",
         "0.0004683483640481878"
        ],
        [
         "75%",
         "0.007570488048572854"
        ],
        [
         "max",
         "0.3098697867522173"
        ]
       ],
       "shape": {
        "columns": 1,
        "rows": 8
       }
      },
      "text/plain": [
       "count    7223.000000\n",
       "mean        0.000362\n",
       "std         0.016464\n",
       "min        -0.163937\n",
       "25%        -0.006719\n",
       "50%         0.000468\n",
       "75%         0.007570\n",
       "max         0.309870\n",
       "Name: Raw_return, dtype: float64"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data_new['1995-01':'2024-09']['Raw_return'].describe()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 分位数 quantile"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "分位数（Quantile），亦称分位点，是指将一个随机变量的概率分布范围分为几个等份的数值点，常用的有中位数（即二分位数）、四分位数、百分位数等。\n",
    "\n",
    "举例说明，我们班上有10名同学，他们的期末成绩是\n",
    "\n",
    "【20，24，38，34，43，56，76，86，98，100】\n",
    "\n",
    "如果期末考试要让40%的不通过，60%的人通过，分数线是多少呢？"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "常见的分类方法：\n",
    "\n",
    "- 中位数 Median\n",
    "  - 当N是奇数时\n",
    "    - $$m_{0.5}=X_{(N+1) / 2}$$\n",
    "  - 当N是偶数时\n",
    "    - $$m_{0.5}=\\frac{X_{(N / 2)}+X_{(N / 2+1)}}{2}$$\n",
    "- 四分位数 Quartile\n",
    "- 10分位数 常用在股票投资策略中"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7223"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(data_new['1995-01':'2024-09'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3903"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(data_new[data_new[\"Raw_return\"] > 0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5242444593687038"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(data_new[data_new[\"Raw_return\"] > 0])/len(data_new)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-0.016149825200764644,\n",
       " -0.008725465267893506,\n",
       " -0.004824058860378355,\n",
       " -0.001823562879049323,\n",
       " 0.000496269119653947,\n",
       " 0.0028404410444615724,\n",
       " 0.005705730958616373,\n",
       " 0.009680619902095611,\n",
       " 0.016938708022350025]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from statistics import quantiles\n",
    "\n",
    "# quantiles函数里面的参数需要注意\n",
    "# 计算data_new在1995-01到2025-08期间Raw_return的十分位数\n",
    "# n=10 表示分成10份（十分位数），method='exclusive' 表示使用不包含端点的分位数计算方法\n",
    "quantiles(data_new['1995-01':'2025-08']['Raw_return'], n = 10, method='exclusive')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-0.016273486796843283,\n",
       " -0.008904480681335824,\n",
       " -0.0049105335359804995,\n",
       " -0.0019112410821794379,\n",
       " 0.0004683483640481878,\n",
       " 0.0028484509548271308,\n",
       " 0.005764948233341815,\n",
       " 0.009789313259894784,\n",
       " 0.017182938215153553]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "quantiles(data_new['1995-01':'2024-09']['Raw_return'], n = 10, method='inclusive') # 在金融里，比如构造分位数组合（5分位、10分位），常用的还是 inclusive（更贴近排序分组）"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on function quantiles in module statistics:\n",
      "\n",
      "quantiles(data, *, n=4, method='exclusive')\n",
      "    Divide *data* into *n* continuous intervals with equal probability.\n",
      "\n",
      "    Returns a list of (n - 1) cut points separating the intervals.\n",
      "\n",
      "    Set *n* to 4 for quartiles (the default).  Set *n* to 10 for deciles.\n",
      "    Set *n* to 100 for percentiles which gives the 99 cuts points that\n",
      "    separate *data* in to 100 equal sized groups.\n",
      "\n",
      "    The *data* can be any iterable containing sample.\n",
      "    The cut points are linearly interpolated between data points.\n",
      "\n",
      "    If *method* is set to *inclusive*, *data* is treated as population\n",
      "    data.  The minimum value is treated as the 0th percentile and the\n",
      "    maximum value is treated as the 100th percentile.\n",
      "\n"
     ]
    }
   ],
   "source": [
    "import statistics\n",
    "help(statistics.quantiles)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[2.75, 5.5, 8.25]\n"
     ]
    }
   ],
   "source": [
    "data = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]\n",
    "q = statistics.quantiles(data, n=4, method='exclusive')\n",
    "print(q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[3.25, 5.5, 7.75]\n"
     ]
    }
   ],
   "source": [
    "q = statistics.quantiles(data, n=4, method='inclusive')\n",
    "print(q)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.microsoft.datawrangler.viewer.v0+json": {
       "columns": [
        {
         "name": "index",
         "rawType": "object",
         "type": "string"
        },
        {
         "name": "Ret",
         "rawType": "float64",
         "type": "float"
        }
       ],
       "ref": "092508c6-ecf8-45d4-8d68-899a9217729f",
       "rows": [
        [
         "count",
         "308.0"
        ],
        [
         "mean",
         "0.005821246663095831"
        ],
        [
         "std",
         "0.06978508810612695"
        ],
        [
         "min",
         "-0.2463139446677537"
        ],
        [
         "25%",
         "-0.03589835517117776"
        ],
        [
         "50%",
         "0.004568893904690174"
        ],
        [
         "75%",
         "0.041992739242666444"
        ],
        [
         "max",
         "0.2744642236184609"
        ]
       ],
       "shape": {
        "columns": 1,
        "rows": 8
       }
      },
      "text/plain": [
       "count    308.000000\n",
       "mean       0.005821\n",
       "std        0.069785\n",
       "min       -0.246314\n",
       "25%       -0.035898\n",
       "50%        0.004569\n",
       "75%        0.041993\n",
       "max        0.274464\n",
       "Name: Ret, dtype: float64"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Month_data['2000-01':'2025-08']['Ret'].describe()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-0.0709964521333589,\n",
       " -0.047737015530115404,\n",
       " -0.026630590375295526,\n",
       " -0.007411841595713531,\n",
       " 0.004471851221674816,\n",
       " 0.020217863557133686,\n",
       " 0.03625761917881416,\n",
       " 0.05352850908377551,\n",
       " 0.09697370519393292]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "statistics.quantiles(Month_data['Ret'], n = 10, method='inclusive')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/var/folders/2c/bzxy6cc102b12rngr422_89h0000gn/T/ipykernel_7277/1150889561.py:1: UserWarning: Boolean Series key will be reindexed to match DataFrame index.\n",
      "  len(Month_data['1995-01':'2024-09'][Month_data[\"Ret\"] > 0])/len(Month_data['1995-01':'2025-08'])\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "0.5108695652173914"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(Month_data['1995-01':'2024-09'][Month_data[\"Ret\"] > 0])/len(Month_data['1995-01':'2025-08'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-0.06407330856971007,\n",
       " -0.04690477004328284,\n",
       " -0.02294678764968542,\n",
       " -0.006030700018074264,\n",
       " 0.004568893904690174,\n",
       " 0.018850177397894186,\n",
       " 0.03537258283387652,\n",
       " 0.05187838634791797,\n",
       " 0.08409364977091105]"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "quantiles(Month_data['2000-01':'2025-08']['Ret'], n = 10, method='exclusive')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0. , 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1. ])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "seq = np.linspace(0,1,11)\n",
    "seq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.08840689, -0.01536682, -0.00833553, -0.00468863, -0.00176721,\n",
       "        0.00049848,  0.00268087,  0.00538043,  0.00914657,  0.01601642,\n",
       "        0.09856839])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.quantile(data_new['2000-01':'2024-09']['Raw_return'],q=seq)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 波动率 Variance"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\sigma^2=\\frac{\\sum(X-\\mu)^2}{N}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ R_{t+1} = a + b * VAR_t $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0002137799963357011"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.var(data_new['2000-01':'2025-08']['Raw_return'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00021377999633570064"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = np.mean(data_new['2000-01':'2025-08']['Raw_return'])\n",
    "sum((data_new['2000-01':'2025-08']['Raw_return'] - m)**2) / len(data_new['2000-01':'2025-08']['Raw_return'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.01462121733426123"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.std(data_new['2000-01':'2025-08']['Raw_return'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(0.01462121733426123)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.sqrt(np.var(data_new['2000-01':'2025-08']['Raw_return']))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.014732982591047728"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from math import sqrt\n",
    "\n",
    "sqrt(np.var(data_new['2000-01':'2024-09']['Raw_return']))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.microsoft.datawrangler.viewer.v0+json": {
       "columns": [
        {
         "name": "month",
         "rawType": "datetime64[ns]",
         "type": "datetime"
        },
        {
         "name": "variance",
         "rawType": "float64",
         "type": "float"
        }
       ],
       "ref": "4a90d95c-2e2d-4339-96c2-c06a2c3aed24",
       "rows": [
        [
         "1995-01-31 00:00:00",
         "0.00026048572657558266"
        ],
        [
         "1995-02-28 00:00:00",
         "0.0011295320824296045"
        ],
        [
         "1995-03-31 00:00:00",
         "0.0005508263417010917"
        ],
        [
         "1995-04-30 00:00:00",
         "0.0004067797543293275"
        ],
        [
         "1995-05-31 00:00:00",
         "0.0069240112632990335"
        ],
        [
         "1995-06-30 00:00:00",
         "0.0004154358290163324"
        ],
        [
         "1995-07-31 00:00:00",
         "0.000561932321189771"
        ],
        [
         "1995-08-31 00:00:00",
         "0.0003142233686155243"
        ],
        [
         "1995-09-30 00:00:00",
         "0.0002059727496917454"
        ],
        [
         "1995-10-31 00:00:00",
         "0.00036092052633565923"
        ],
        [
         "1995-11-30 00:00:00",
         "0.00014365283808199554"
        ],
        [
         "1995-12-31 00:00:00",
         "0.00017225591483781397"
        ],
        [
         "1996-01-31 00:00:00",
         "0.0003657480853493107"
        ],
        [
         "1996-02-29 00:00:00",
         "0.00016535212171368545"
        ],
        [
         "1996-03-31 00:00:00",
         "0.0006122452737349079"
        ],
        [
         "1996-04-30 00:00:00",
         "0.0009994813644346578"
        ],
        [
         "1996-05-31 00:00:00",
         "0.0005421148188997528"
        ],
        [
         "1996-06-30 00:00:00",
         "0.0006459671113744179"
        ],
        [
         "1996-07-31 00:00:00",
         "0.0006612321146007734"
        ],
        [
         "1996-08-31 00:00:00",
         "0.0002766829834219439"
        ],
        [
         "1996-09-30 00:00:00",
         "0.0005441376441252792"
        ],
        [
         "1996-10-31 00:00:00",
         "0.0004308197268154401"
        ],
        [
         "1996-11-30 00:00:00",
         "0.0007572987449973056"
        ],
        [
         "1996-12-31 00:00:00",
         "0.0025884901614212107"
        ],
        [
         "1997-01-31 00:00:00",
         "0.00016012032719245814"
        ],
        [
         "1997-02-28 00:00:00",
         "0.0018646771804899712"
        ],
        [
         "1997-03-31 00:00:00",
         "0.0003019954103282529"
        ],
        [
         "1997-04-30 00:00:00",
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       "            variance\n",
       "month               \n",
       "1995-01-31  0.000260\n",
       "1995-02-28  0.001130\n",
       "1995-03-31  0.000551\n",
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       "1995-05-31  0.006924\n",
       "...              ...\n",
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       "\n",
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    "monthly_var = data_new.resample('ME')['Raw_return'].var().to_frame(name = 'variance')\n",
    "monthly_var.index.name = 'month'\n",
    "monthly_var"
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       "      <th>1995-04-30</th>\n",
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       "            variance\n",
       "month               \n",
       "1995-01-31  0.005695\n",
       "1995-02-28  0.018086\n",
       "1995-03-31  0.013378\n",
       "1995-04-30  0.008281\n",
       "1995-05-31  0.148387\n",
       "...              ...\n",
       "2025-04-30  0.006097\n",
       "2025-05-31  0.000602\n",
       "2025-06-30  0.000609\n",
       "2025-07-31  0.000464\n",
       "2025-08-31  0.001411\n",
       "\n",
       "[368 rows x 1 columns]"
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     "execution_count": 36,
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   "source": [
    "monthly_var = data_new.resample('ME')['Raw_return'].apply(lambda x: np.sum(x ** 2)).to_frame(name = 'variance')\n",
    "monthly_var.index.name = 'month'\n",
    "monthly_var"
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 正态分布（Normal Distribution）"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "若随机变量X服从一个数学期望为 $\\mu$ 、方差为 $\\sigma^{2}$ 的正态分布，记为 $\\mathrm{N}\\left(\\mu, \\sigma^{2}\\right)$ 。其概率密度函数为正态分布的期望值决定了 其位置，其标准差$\\sigma$决定了分布的幅度。当 $\\mu=0, \\sigma=1$ 时的正态分布是标准正态分布。\n",
    "\n",
    "若随机变量 $X$ 服从一个位置参数为 $\\mu$ 、尺度参数为 $\\sigma$ 的概率分布，且其概率密度函数为 \n",
    "$$\n",
    "f(x)=\\frac{1}{\\sqrt{2 \\pi} \\sigma} \\exp \\left(-\\frac{(x-\\mu)^{2}}{2 \\sigma^{2}}\\right)\n",
    "$$\n",
    "则这个随机变量就称为正态随机变量, 正态随机变量服从的分布就称为正态分布, 记作 $X \\sim N\\left(\\mu, \\sigma^{2}\\right)$, 读作 $X$ 服从 $N\\left(\\mu, \\sigma^{2}\\right)$, 或 $X$ 服从正态分布。\n",
    "\n",
    " 若 $X \\sim N\\left(\\mu, \\sigma^{2}\\right), Y=\\frac{X-\\mu}{\\sigma} \\sim N(0,1)$.\n",
    " \n",
    " 当 $\\mu=0, \\sigma=1$ 时，正态分布就成为标准正态分布\n",
    "$$\n",
    "f(x)=\\frac{1}{\\sqrt{2 \\pi}} e^{\\left(-\\frac{x^{2}}{2}\\right)}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 偏度和峰度"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 偏度（Skewness）是统计数据分布偏斜方向和程度的度量，是统计数据分布非对称程度的数字特征。定义上偏度是样本的三阶标准化矩。\n",
    "$$\n",
    "S=\\frac{1}{n} \\sum_{i=1}^{n}\\left[\\left(\\frac{X_{i}-\\mu}{\\sigma}\\right)^{3}\\right]\n",
    "$$\n",
    "\n",
    "其中 $\\mu$ 是均值， $\\sigma$ 是标准差。定义中包括正态分布（偏度=0），右偏分布（正偏，>0）,左偏分布（负偏，<0）"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 600x900 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.stats import skewnorm, skew\n",
    "import seaborn as sns\n",
    "\n",
    "try:\n",
    "    plt.rcParams['font.sans-serif'] = ['PingFang SC', 'Arial Unicode MS'] \n",
    "except Exception as e:\n",
    "    print(f\"设置中文字体失败，请检查字体是否安装: {e}\")\n",
    "  \n",
    "\n",
    "np.random.seed(42) \n",
    "\n",
    "\n",
    "data_neg = skewnorm.rvs(a=-5, loc=20, scale=5, size=n_samples)\n",
    "skew_neg = skew(data_neg)\n",
    "\n",
    "\n",
    "data_zero = skewnorm.rvs(a=0, loc=0, scale=5, size=n_samples)\n",
    "skew_zero = skew(data_zero)\n",
    "\n",
    "\n",
    "data_pos = skewnorm.rvs(a=5, loc=-20, scale=5, size=n_samples)\n",
    "skew_pos = skew(data_pos)\n",
    "\n",
    "\n",
    "fig, axes = plt.subplots(3,1, figsize=(6, 9))\n",
    "fig.suptitle('偏度分布示例', fontsize=20)\n",
    "\n",
    "\n",
    "\n",
    "sns.histplot(data_neg, kde=True, ax=axes[0], color='coral', bins=50)\n",
    "axes[0].set_title(f'负偏度 (左偏)\\n偏度值 ≈ {skew_neg:.2f}', fontsize=16)\n",
    "axes[0].set_xlabel('数值', fontsize=12)\n",
    "axes[0].set_ylabel('频数', fontsize=12)\n",
    "\n",
    "mean_neg = np.mean(data_neg)\n",
    "median_neg = np.median(data_neg)\n",
    "axes[0].axvline(mean_neg, color='red', linestyle='--', linewidth=2, label=f'均值: {mean_neg:.2f}')\n",
    "axes[0].axvline(median_neg, color='green', linestyle='-', linewidth=2, label=f'中位数: {median_neg:.2f}')\n",
    "axes[0].legend()\n",
    "axes[0].text(0.05, 0.9, '均值 < 中位数', transform=axes[0].transAxes, fontsize=12, color='blue')\n",
    "\n",
    "\n",
    "\n",
    "sns.histplot(data_zero, kde=True, ax=axes[1], color='skyblue', bins=50)\n",
    "axes[1].set_title(f'零偏度 (对称)\\n偏度值 ≈ {skew_zero:.2f}', fontsize=16)\n",
    "axes[1].set_xlabel('数值', fontsize=12)\n",
    "axes[1].set_ylabel('') \n",
    "mean_zero = np.mean(data_zero)\n",
    "median_zero = np.median(data_zero)\n",
    "axes[1].axvline(mean_zero, color='red', linestyle='--', linewidth=2, label=f'均值: {mean_zero:.2f}')\n",
    "axes[1].axvline(median_zero, color='green', linestyle='-', linewidth=2, label=f'中位数: {median_zero:.2f}')\n",
    "axes[1].legend()\n",
    "axes[1].text(0.05, 0.9, '均值 ≈ 中位数', transform=axes[1].transAxes, fontsize=12, color='blue')\n",
    "\n",
    "\n",
    "\n",
    "sns.histplot(data_pos, kde=True, ax=axes[2], color='lightgreen', bins=50)\n",
    "axes[2].set_title(f'正偏度 (右偏)\\n偏度值 ≈ {skew_pos:.2f}', fontsize=16)\n",
    "axes[2].set_xlabel('数值', fontsize=12)\n",
    "axes[2].set_ylabel('')\n",
    "mean_pos = np.mean(data_pos)\n",
    "median_pos = np.median(data_pos)\n",
    "axes[2].axvline(mean_pos, color='red', linestyle='--', linewidth=2, label=f'均值: {mean_pos:.2f}')\n",
    "axes[2].axvline(median_pos, color='green', linestyle='-', linewidth=2, label=f'中位数: {median_pos:.2f}')\n",
    "axes[2].legend()\n",
    "axes[2].text(0.05, 0.9, '均值 > 中位数', transform=axes[2].transAxes, fontsize=12, color='blue')\n",
    "\n",
    "\n",
    "\n",
    "plt.tight_layout(rect=[0, 0, 1, 0.95]) \n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* 峰度（Kurtosis）表征概率密度分布曲线在平均值处峰度高低的特征。直观来看，峰度反应了峰部的尖度。其计算方法为\n",
    "   $$K=\\frac{1}{n} \\sum_{i=1}^{n}\\left[\\left(\\frac{X_{i}-\\mu}{\\sigma}\\right)^{4}\\right]$$\n",
    "    其中 $\\mu$ 是均值， $\\sigma$ 是标准差。定义中包括正态分布（峰度=3），厚尾（峰度>3）,瘦尾（峰度<3）。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
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       "     <use xlink:href=\"#ArialUnicodeMS-6b\" transform=\"translate(811.181641 0)\"/>\n",
       "     <use xlink:href=\"#ArialUnicodeMS-75\" transform=\"translate(861.181641 0)\"/>\n",
       "     <use xlink:href=\"#ArialUnicodeMS-72\" transform=\"translate(916.796875 0)\"/>\n",
       "     <use xlink:href=\"#ArialUnicodeMS-74\" transform=\"translate(950.097656 0)\"/>\n",
       "     <use xlink:href=\"#ArialUnicodeMS-69\" transform=\"translate(977.880859 0)\"/>\n",
       "     <use xlink:href=\"#ArialUnicodeMS-63\" transform=\"translate(1000.097656 0)\"/>\n",
       "     <use xlink:href=\"#ArialUnicodeMS-29\" transform=\"translate(1050.097656 0)\"/>\n",
       "    </g>\n",
       "    <!-- 峰度值 ≈ 1.81 (&lt; 3) -->\n",
       "    <g transform=\"translate(165.504375 483.8415) scale(0.16 -0.16)\">\n",
       "     <defs>\n",
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       "\" transform=\"scale(0.015625)\"/>\n",
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       "  </g>\n",
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       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 600x900 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.stats import kurtosis, norm, laplace, uniform\n",
    "import seaborn as sns\n",
    "\n",
    "\n",
    "try:\n",
    "    plt.rcParams['font.sans-serif'] = ['PingFang SC', 'Arial Unicode MS'] \n",
    "    plt.rcParams['axes.unicode_minus'] = False \n",
    "except Exception as e:\n",
    "    print(f\"设置中文字体失败，请检查字体是否安装: {e}\")\n",
    "   \n",
    "\n",
    "\n",
    "np.random.seed(42)\n",
    "n_samples = 10000\n",
    "\n",
    "\n",
    "data_lepto = laplace.rvs(loc=0, scale=1, size=n_samples)\n",
    "\n",
    "kurt_lepto = kurtosis(data_lepto, fisher=False)\n",
    "\n",
    "\n",
    "data_meso = norm.rvs(loc=0, scale=1.5, size=n_samples) \n",
    "kurt_meso = kurtosis(data_meso, fisher=False)\n",
    "\n",
    "\n",
    "data_platy = uniform.rvs(loc=-3, scale=6, size=n_samples)\n",
    "kurt_platy = kurtosis(data_platy, fisher=False)\n",
    "\n",
    "\n",
    "fig, axes = plt.subplots(3, 1, figsize=(6, 9))\n",
    "fig.suptitle('峰度分布示例', fontsize=20)\n",
    "common_xlim = (-6, 6) \n",
    "\n",
    "\n",
    "sns.histplot(data_lepto, kde=True, ax=axes[0], color='tomato', bins=50)\n",
    "axes[0].set_title(f'尖峰/厚尾 (Leptokurtic)\\n峰度值 ≈ {kurt_lepto:.2f} (> 3)', fontsize=16)\n",
    "axes[0].set_xlabel('数值', fontsize=12)\n",
    "axes[0].set_ylabel('频数', fontsize=12)\n",
    "axes[0].set_xlim(common_xlim)\n",
    "\n",
    "\n",
    "sns.histplot(data_meso, kde=True, ax=axes[1], color='skyblue', bins=50)\n",
    "axes[1].set_title(f'常态峰 (Mesokurtic)\\n峰度值 ≈ {kurt_meso:.2f} (≈ 3)', fontsize=16)\n",
    "axes[1].set_xlabel('数值', fontsize=12)\n",
    "axes[1].set_ylabel('')\n",
    "axes[1].set_xlim(common_xlim)\n",
    "\n",
    "\n",
    "sns.histplot(data_platy, kde=True, ax=axes[2], color='lightgreen', bins=50)\n",
    "axes[2].set_title(f'低阔峰/瘦尾 (Platykurtic)\\n峰度值 ≈ {kurt_platy:.2f} (< 3)', fontsize=16)\n",
    "axes[2].set_xlabel('数值', fontsize=12)\n",
    "axes[2].set_ylabel('')\n",
    "axes[2].set_xlim(common_xlim)\n",
    "\n",
    "\n",
    "plt.tight_layout(rect=[0, 0, 1, 0.95])\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7445"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "np.float64(0.6772796464034425)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "np.float64(23.63802106137419)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "np.float64(23.621343146464902)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "np.float64(23.621343146464902)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "len(data_new['1995-01':'2025-08'])\n",
    "data_new['1995-01':'2025-08']['Raw_return'].skew()\n",
    "data_new['1995-01':'2025-08']['Raw_return'].kurt()\n",
    "scipy.stats.kurtosis(data_new['1995-01':'2025-08']['Raw_return'],fisher=False) - 3\n",
    "scipy.stats.kurtosis(data_new['1995-01':'2025-08']['Raw_return'],fisher=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "23.45514481569284"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = np.mean(data_new['1995-01':'2024-09']['Raw_return'])\n",
    "l = len(data_new['1995-01':'2024-09']['Raw_return'])\n",
    "sum(((data_new['1995-01':'2024-09']['Raw_return'] - m)/np.std(data_new['1995-01':'2024-09']['Raw_return']))**4) / l -3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(0.32934252744736264)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "np.float64(1.9091672959136194)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Monthly data\n",
    "Month_data['1995-01':'2024-09']['Ret'].skew()\n",
    "Month_data['1995-01':'2024-09']['Ret'].kurt()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(0.8675024485714982)"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "np.float64(1.7137615374448099)"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Quarterly data\n",
    "Quarter_data['1995-01':'2024-09']['Ret'].skew()\n",
    "Quarter_data['1995-01':'2024-09']['Ret'].kurt()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.1582469738165153"
      ]
     },
     "execution_count": 142,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "1.6143563245668822"
      ]
     },
     "execution_count": 142,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Year_data['1995-01':'2024-09']['Raw_return'].skew()\n",
    "Year_data['1995-01':'2024-09']['Raw_return'].kurt()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 标准正态分布的直方图"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
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       "    <dc:creator>\n",
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   "source": [
    "\n",
    "\n",
    "'''\"\n",
    "normfun(x, mu, sigma) 函数定义了一个正态分布的概率密度函数，\n",
    "其中 x 是自变量\n",
    "mu 是均值\n",
    "sigma 是标准差\n",
    "这个函数计算了在给定均值和标准差下 x 处的概率密度值。\n",
    "'''\n",
    "\n",
    "def normfun(x, mu, sigma):\n",
    "  pdf = np.exp(-((x - mu)**2)/(2*sigma**2)) / (sigma * np.sqrt(2*np.pi))\n",
    "  return pdf\n",
    "\n",
    "\n",
    "\n",
    "result = np.random.normal(0, 1, 1000000) # mean 0 and standard deviation 1\n",
    "\n",
    "\n",
    "\n",
    "x = np.arange(min(result), max(result),0.01)\n",
    "\n",
    "\n",
    "y = normfun(x, result.mean(), result.std())\n",
    "\n",
    "\n",
    "plt.figure(figsize=(8, 4)) \n",
    "plt.plot(x, y) \n",
    "\n",
    "fig = plt.hist(result, bins=100, rwidth=1, density=True) \n",
    "plt.title('distribution')\n",
    "#plt.xlabel('')\n",
    "plt.ylabel('probability')\n",
    "# 输出\n",
    "plt.show(); "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 在日收益率直方图上添加正态分布密度曲线"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 1000x500 with 1 Axes>"
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   "source": [
    "retmean = data_new['2000-01':'2025-08']['Raw_return'].mean()\n",
    "retstd = data_new['2000-01':'2025-08']['Raw_return'].std()\n",
    "retmin = data_new['2000-01':'2025-08']['Raw_return'].min()\n",
    "retmax = data_new['2000-01':'2025-08']['Raw_return'].max()\n",
    "\n",
    "\n",
    "def normfun(x, mu, sigma):\n",
    "    pdf = np.exp(-((x - mu)**2) /\n",
    "                 (2 * sigma**2)) / (sigma * np.sqrt(2 * np.pi))\n",
    "    return pdf\n",
    "\n",
    "\n",
    "x = np.arange(retmin, retmax, 0.001) # 0.001 0.002 0.003 0.004.。。。 0.099 0.100  0.101\n",
    "y = normfun(x, retmean, retstd)\n",
    "fig = plt.figure(figsize=(10, 5))\n",
    "plt.plot(x, y)\n",
    "\n",
    "fig = plt.hist(data_new['2000-01':'2025-08']['Raw_return'], # index daily market excess return\n",
    "               bins=1000,\n",
    "               histtype='bar',\n",
    "               color='r',\n",
    "               alpha=0.9)\n",
    "plt.xticks(rotation=90, fontsize=8) # x轴显示方法\n",
    "plt.title('Distribution')\n",
    "plt.xlabel('Market Return From January 2000 to Augest 2025')\n",
    "plt.ylabel('Probability')\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 在月收益率直方图上添加正态分布密"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
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      ],
      "text/plain": [
       "<Figure size 1000x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "retmean = Month_data['2000-01':'2025-08']['Ret'].mean()\n",
    "retstd = Month_data['2000-01':'2025-08']['Ret'].std()\n",
    "retmin = Month_data['2000-01':'2025-08']['Ret'].min()\n",
    "retmax = Month_data['2000-01':'2025-08']['Ret'].max()\n",
    "\n",
    "\n",
    "def normfun(x, mu, sigma):\n",
    "    pdf = np.exp(-((x - mu)**2) /\n",
    "                 (2 * sigma**2)) / (sigma * np.sqrt(2 * np.pi))\n",
    "    return pdf\n",
    "\n",
    "\n",
    "x = np.arange(retmin, retmax, 0.001)\n",
    "y = normfun(x, retmean, retstd)\n",
    "fig = plt.figure(figsize=(10, 5)) # 图片大小 10，5\n",
    "plt.plot(x, y)\n",
    "\n",
    "fig = plt.hist(Month_data['2000-01':'2025-08']['Ret'],\n",
    "               bins=100,\n",
    "               histtype='bar',\n",
    "               color='r',\n",
    "               alpha=0.9)\n",
    "plt.xticks(rotation=90, fontsize=8)\n",
    "plt.title('Distribution')\n",
    "plt.xlabel('Market Return From January 2000 to December 2023')\n",
    "plt.ylabel('Probability')\n",
    "plt.show();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 平稳性 stationary\n",
    "\n",
    "\n",
    "### 什么是平稳性 (Stationarity)？\n",
    "\n",
    "想象一下，你正在观察一条河流。如果你在任何一天、任何一周、任何一年去观察，发现这条河的水位总是在一个相似的高度上下波动，水流的速度也维持在一个稳定的范围内，那么我们可以说这条河的状貌是“平稳”的。\n",
    "\n",
    "在时间序列分析中，“平稳性”就是类似的概念。一个时间序列如果具有平稳性，意味着它的统计特性不随时间的变化而变化。具体来说，我们通常关注的是**弱平稳性**（也叫协方差平稳性），它要求满足以下三个条件：\n",
    "\n",
    "1.  **均值恒定**：序列的均值（期望值）是一个常数，不随时间t的变化而改变。就像那条河，无论你在1月份还是7月份测量，它的平均水位都是相似的。\n",
    "2.  **方差恒定**：序列的方差是一个常数，不随时间t的变化而改变。这意味着序列的波动程度是稳定的。河水的波动幅度（浪花的大小）不会因为季节变化而发生系统性的剧烈改变。\n",
    "3.  **协方差只与时间间隔有关**：序列在任意两个时间点（t和t-k）的协方差，只取决于它们之间的时间间隔k，而与具体的时间点t无关。也就是说，今天和昨天的关联程度，与去年今天和去年昨天的关联程度是一样的。\n",
    "\n",
    "**一个非平稳的例子**：一支股票的价格。比如S&P 500指数，从1980年到今天，它的均值（平均价格水平）明显是不断上升的，这违背了“均值恒定”的原则，所以股价序列通常是非平稳的。\n",
    "\n",
    "### 什么是遍历性 (Ergodicity)？\n",
    "\n",
    "遍历性是一个更抽象，但同样重要的概念。它关系到我们如何利用有限的数据去推断总体的规律。\n",
    "\n",
    "**一个经典的类比**：假设我们想知道一个赌场里所有**同款老虎机**的平均中奖概率。我们有两种方法：\n",
    "1.  **空间平均 (Ensemble Average)**：在某个瞬间，把赌场里1000台老虎机全部玩一遍，然后计算这1000次结果的平均值。\n",
    "2.  **时间平均 (Time Average)**：只选择**其中一台**老虎机，连续不断地玩1000次，然后计算这1000次结果的平均值。\n",
    "\n",
    "如果这两种方法计算出的平均中奖概率是相等的，那么我们就说这个系统（所有老虎机的集合）具有**遍历性**。\n",
    "\n",
    "在金融领域，我们面对的现实是：我们永远无法进行“空间平均”。我们观察不到S&P 500指数在“平行宇宙”中的其他可能路径。我们拥有的，仅仅是它从诞生至今的**一条**历史路径数据。遍历性假设允许我们相信，通过分析这一条足够长的时间序列（时间平均），我们就可以推断出这个指数背后数据生成过程的真实统计特性（空间平均）。\n",
    "\n",
    "简单来说，**平稳性是遍历性的前提条件**。一个过程如果连自身的统计特性都在随时间变化（非平稳），那么我们观察的任何一段历史都不能代表它的“全部”，时间平均也就无从谈起了。\n",
    "\n",
    "### 为什么平稳性在金融时间序列中如此重要？\n",
    "\n",
    "平稳性是金融时间序列分析的基石，重要性体现在以下几个方面：\n",
    "\n",
    "#### 1. 它是模型有效性的前提\n",
    "许多经典的计量经济学模型，比如自回归模型（AR）、移动平均模型（MA）以及ARMA模型，都要求输入的数据是平稳的。这些模型的核心思想是，利用序列过去的行为和误差项来预测未来。如果序列的均值和方差等统计特性一直在变，那么基于历史数据估计出的模型参数就会不稳定，甚至毫无意义，无法用于未来的预测。\n",
    "\n",
    "#### 2. 它是有效预测的基础\n",
    "我们的目标是预测未来。如果一个时间序列是非平稳的，意味着它的行为模式在不断改变。比如一个公司的销售额，如果存在持续的增长趋势（均值不断变大），那么用过去十年的平均销售额来预测下个月的销售额，显然是荒谬的。只有当序列平稳时，我们才能假设“历史会在一定程度上重演”，过去的统计规律（如均值、波动性）在未来依然适用，从而让我们的预测有据可依。\n",
    "\n",
    "#### 3. 它可以帮助我们避免“伪回归”（Spurious Regression）的陷阱\n",
    "这是金融实证研究中最著名、也最危险的陷阱之一，也是对本科生最直观的警示。\n",
    "\n",
    "**真实的金融事实说明**：\n",
    "想象一下，你收集了两个毫不相关的时间序列数据：1990年到2020年“美国的名义GDP”和“马来西亚的棕榈油产量”。这两个序列很可能都是非平稳的，因为它们都随着经济发展和技术进步表现出明显的上升趋势。\n",
    "\n",
    "如果你直接将这两个非平稳序列进行回归分析，比如 `GDP = a + b * PalmOil + e`，你很可能会得到一个**极其“漂亮”**的结果：\n",
    "*   **很高的R²值**（比如0.9以上），显示模型解释力超强。\n",
    "*   **非常显著的t统计量**，显示棕榈油产量对GDP有显著的解释能力。\n",
    "\n",
    "看到这样的结果，你可能会兴奋地得出结论：“马来西亚的棕榈油产量是驱动美国GDP增长的关键因素！” 这显然是荒谬的。这种看似显著的统计关系其实是虚假的，它仅仅是因为两个序列共享了相似的时间趋势（都在增长），而并非存在任何真实的经济联系。这就是“伪回归”。\n",
    "\n",
    "**如何解决？**\n",
    "这个问题的根源在于序列的非平稳性。在金融实践中，我们通常不会直接对股价或GDP这样的水平值（Level）建模。我们会先对其进行**差分（Differencing）**处理，将其转换为平稳序列。\n",
    "\n",
    "*   **股价 vs. 股票收益率**：股价（Price）是非平稳的，但它的对数差分，也就是我们常说的**对数收益率**（Log Return），通常是平稳的。股票收益率序列的均值会稳定地围绕一个小的正数波动，其方差也相对稳定。\n",
    "*   **GDP vs. GDP增长率**：GDP总量是非平稳的，但GDP的同比增长率或环比增长率通常是平稳的。\n",
    "\n",
    "通过对数据进行差分等变换，我们把非平稳的序列变成了平稳的序列。然后，我们再用平稳的收益率或增长率序列去构建模型和进行回归分析，这样得出的结论才是统计上可靠的，避免了伪回归的谬误。\n",
    "\n",
    "**总结一下**：\n",
    "你可以这样理解：平稳性就像是为我们的金融数据分析设定了“公平的赛道”。如果数据本身在随时间“变形”（非平稳），那么任何在历史赛道上总结出的规律都无法指导未来的比赛。我们必须通过差分等手段，将数据“拉直”到一条平稳的赛道上，然后才能运用各种强大的计量工具进行建模、预测和检验，得出有意义的结论。遍历性则为我们提供了理论上的信心，让我们相信在平稳的赛道上，跑得足够久，就能看清赛道本身的全貌。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "原始序列的ADF检验结果:\n",
      "ADF Statistic: -6.0735\n",
      "p-value: 0.0000\n",
      "Critical Values:\n",
      "   1%: -3.4537\n",
      "   5%: -2.8718\n",
      "   10%: -2.5722\n",
      "结论: p-value小于0.05，拒绝原假设，序列是平稳的。\n"
     ]
    }
   ],
   "source": [
    "from statsmodels.tsa.stattools import adfuller as ADF\n",
    "\n",
    "# 对月收益率数据进行ADF检验\n",
    "adf_result = ADF(Month_data['2000-01':'2024-09']['Ret'])\n",
    "\n",
    "print('原始序列的ADF检验结果:')\n",
    "print(f'ADF Statistic: {adf_result[0]:.4f}')\n",
    "print(f'p-value: {adf_result[1]:.4f}')\n",
    "print('Critical Values:')\n",
    "for key, value in adf_result[4].items():\n",
    "    print(f'   {key}: {value:.4f}')\n",
    "\n",
    "if adf_result[1] <= 0.05:\n",
    "    print('结论: p-value小于0.05，拒绝原假设，序列是平稳的。')\n",
    "else:\n",
    "    print('结论: p-value大于0.05，未能拒绝原假设，序列是非平稳的。')"
   ]
  }
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